How do you determine the stability of a numerical algorithm?

To determine the stability of a numerical algorithm, analyzing error propagation is essential. Stability in numerical methods refers to how errors are handled throughout calculations, particularly how small perturbations or errors in the input can influence the final output. If an algorithm is stable, it ensures that any errors do not grow uncontrollably as computations proceed. This means even if a small error is introduced at the initial stage of calculations, the resulting output will remain close to the true value, which is crucial for the reliability and accuracy of numerical solutions.

For example, one way to assess stability is by examining how the error changes with different input sizes or after multiple iterations. Algorithms that exhibit stability will typically show that errors remain bounded or decline as the computation proceeds, whereas unstable algorithms might allow errors to escalate, leading to significant deviations from the expected outcomes.

In contrast, simply checking for the existence of a complex root does not provide a complete picture of an algorithm's stability, as stability encompasses more than just the presence of roots. Ensuring calculations are analytically evident is more related to the transparency and reproducibility of the calculations rather than their stability. Finally, while comparing outputs to previous results can provide some insight into the behavior of the algorithm, it does not directly assess how errors